From Bucket-Elimination To Bucket Trees E Bucket E: P(E|B,C) - - PDF document

Bucket E: P(E|B,C) Bucket D: P(D|A,B) Bucket C: P(C|A) Bucket B: P(B|A) Bucket A: P(A)

) , ( C B

C E

λ

P(E|B,C) P(C|A) P(A) P(B|A) P(D|A,B)

) , ( C B λ ) , ( B A λ ) , ( B A λ

) (A

B

λ

) , ( B A

B D

λ ) , ( B A

B C

λ

) (A

B

λ

E D C A B

A E D C B

B,C

E

D,B,A

D

B,A

B

A

A

A,B,C

C

T= From Bucket-Elimination To Bucket Trees Definition: T is a bucket tree. Theorem: T is an i-map of G.

• Variable-elimination can be viewed as

message-passing (elimination) using a rooted bucket tree.

• Any variable (bucket) can be the root.

Generalization:Eliminate (sum over) Variables Not in Separators

• Multiply all incoming messages, and Pi’s in

the bucket and sum over B1∩B2.

• Given a rooted bucket tree, T, every node can

be the “root” of the variables-elimination computation.

• If B3 is the root, bucket B2 and then Bucket B1

should be processed; π-messages sent from B2 to B1 and from B1 To B3

) ( ) ( ) (

1 2 1

i i s B B B

P s Π Π ∑ =

• −

λ λ

B4 B3 Pi λ1 λ2 B1 S S=B1∩B2 B2

Bucket Propagation Algorithm

• Input: A bucket tree B1…Bn
• Output: For Each Bi and parent Bj, functions

λi

j(Sij) and πi j (Sij) are exchanged.

Bk Bi λ1 λ Bj

Π

ij j i

S B B = ∩

Top Down:

• Let s λ1… λk messages from child nodes of Bi,

P1…Pr in Bi original functions.

• Bottom Up:
• Let πi

j be received from Bj.

• j

j i i B B ij j i

P S

j i

Π

• Π

= ∑

−

λ λ ) (

i k i i j j j B B ki k i

P S

i k

λ π π Π

• Π

=

≠ −

∑

) ( ) (

• The belief of Bi
• P(Bi) =
• if x index Bucket i

get Bel(x) by summing out Bel(x) = α

π λ

i j i i j i P

• Π
• Π

) (

i S

B P

ij

∑

Propagation in a Bucket Tree Definitions:

• Let G be a Bayesian network, d, an ordering

and B1…Bn the final bucket created processing along d = x1…xn.

• Let Bi be the set of variables appearing in

bucket i when it is processed. Bucket Tree:

• A bucket tree has each Bi cluster as a node and

there is an arc from Bi to Bj if the function created at Bi was placed in Bj Graph-Based Definition:

• Let Gd be the induced graph along d. Each

variable x and it’s earlier neighbors in a node,

• Bx. There is an arc from Bx to By if y is the

closest parent of x.

Upwards Messages On The Bucket Tree

E,B,C A,B,C A B,A A,B,D

) , ( C B λ ) , ( B A λ

) , ( B A λ ) , ( B A

B

Π E D C A B

) (A λ

Π

) , ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( ) , ( ) , ( ) , ( ) ( ) ( B A A C A P C B B A A A B P B A B A A B P B A A P A

C B E C B D C B B C P B

Π

• ∑

= Π

• Π
• =

Π

• =

Π = Π λ λ